Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}8x-2y &= -1 \\ -3x+y &= 4\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $2$ $\begin{align*}8x-2y &= -1\\ -6x+2y &= 8\end{align*}$ Add the top and bottom equations. $2x = 7$ Divide both sides by $2$ and reduce as necessary. $x = \dfrac{7}{2}$ Substitute $\dfrac{7}{2}$ for $x$ in the top equation. $8( \dfrac{7}{2})-2y = -1$ $28-2y = -1$ $-2y = -29$ $y = \dfrac{29}{2}$ The solution is $\enspace x = \dfrac{7}{2}, \enspace y = \dfrac{29}{2}$.